Quadrotor unmanned aerial vehicles (UAVs) have attracted substantial attention due to their simple structure and strong adaptability, but enhancing robustness and adaptability for trajectory tracking under unbounded disturbances remains a key challenge. To address this, this article proposes a novel double integral fuzzy zeroing neural dynamics controller (DIFZNDC). The DIFZNDC integrates a double integral neural dynamics model with a novel dual-input single-output fuzzy logic system (DISOFLS), which can adaptively adjust the parameters, thereby enhancing the robustness and adaptability of the controller. In addition, the global convergence and robustness of the system under the DIFZNDC are theoretically verified. Moreover, two trajectory tracking examples demonstrate the effectiveness and superiority of the DIFZNDC for the quadrotor system. Quantitative analysis under Gaussian disturbance indicates that the DIFZNDC reduces the root-mean-square error (RMSE) by 84.61% and 48.35% compared to the modified super-twisting controller (MSTC) and the fixed-time zeroing neural dynamics controller (FTZNDC), respectively.

This article proposes predefined-time adaptive neural network (PTANN) and event-triggered PTANN (ET-PTANN) models to efficiently compute the time-varying tensor Moore-Penrose (MP) inverse. The PTANN model incorporates a novel adaptive parameter and activation function, enabling it to achieve strongly predefined-time convergence. Unlike traditional time-varying parameters that increase over time, the adaptive parameter is proportional to the error norm, thereby better allocating computational resources and improving efficiency. To further enhance efficiency, the ET-PTANN model combines an event trigger with the evolution formula, resulting in the adjustment of step size and reduction of computation frequency compared to the PTANN model. By conducting mathematical derivations, the article derives the upper bound of convergence time for the proposed neural network models and determines the minimum execution interval for the event trigger. A simulation example demonstrates that the PTANN and ET-PTANN models outperform other related neural network models in terms of computational efficiency and convergence rate. Finally, the practicality of the PTANN and ET-PTANN models is demonstrated through their application for mobile sound source localization.

Time-varying linear equations (TVLEs) play a fundamental role in the engineering field and are of great practical value. Existing methods for the TVLE still have issues with long computation time and insufficient noise resistance. Zeroing neural network (ZNN) with parallel distribution and interference tolerance traits can mitigate these deficiencies and thus are good candidates for the TVLE. Therefore, a new predefined-time adaptive ZNN (PTAZNN) model is proposed for addressing the TVLE in this article. Unlike previous ZNN models with time-varying parameters, the PTAZNN model adopts a novel error-based adaptive parameter, which makes the convergence process more rapid and avoids unnecessary waste of computational resources caused by large parameters. Moreover, the stability, convergence, and robustness of the PTAZNN model are rigorously analyzed. Two numerical examples reflect that the PTAZNN model possesses shorter convergence time and better robustness compared with several variable-parameter ZNN models. In addition, the PTAZNN model is applied to solve the inverse kinematic solution of UR5 robot on the simulation platform CoppeliaSim, and the results further indicate the feasibility of this model intuitively.

The time-varying Lyapunov equation (TVLE) plays a crucial role in control design and system stability. However, there has been limited research conducted on the time-varying generalized Lyapunov equation in the quaternion field. To tackle the time-varying quaternion generalized Lyapunov equation, a nonlinear noise-resistant zeroing neural network (NNR-ZNN) model with a novel power activation function (NPAF) is devised. The issue of non-commutativity within quaternion is circumvented by utilizing the real representation. The theoretical analyses provide a sufficient explanation for the global stability, fixed-time convergence, and robustness of the NNR-ZNN model. Under several different kinds of noises, the exceptional robustness of the NNR-ZNN model is highlighted by comparison with other existing models. In the end, the successful applications of the NNR-ZNN model to color image fusion and color image denoising confirm the practical value of the NNR-ZNN model.

Sliding mode control (SMC) is considered an efficacious scheme for quadrotor control. However, the control performance of the existing SMC schemes depends on initial states and multiple parameters, and the robustness needs to be improved. To address these issues, a novel predefined-time robust SMC framework based on two zeroing neural dynamics (ZND) schemes, referred to as ZND-based predefined-time robust SMC (ZNDPRSMC) framework, is developed to facilitate position and attitude tracking of a quadrotor under bounded disturbances. Initially, a nonsingular sliding mode surface (SMS) is formulated by incorporating a general ZND along with a differentiable predefined-time activation function. Following this, an approaching law is introduced by utilizing a variable-parameter noise-tolerant ZND and a novel dynamic adaptive parameter. The nonsingular SMS and the approaching law are then combined to construct a nonsingular predefined-time robust controller. The theoretical proofs provided ascertain the predefined-time convergence of the closed-loop system utilizing ZNDPRSMC and its robustness against bounded disturbances. Finally, two trajectory tracking examples of the quadrotor are presented to demonstrate the superiority of the ZNDPRSMC framework.

In this article, a novel adaptive robust predefined-time nonsingular sliding mode control (ARPTNSMC) scheme is investigated, which aims to achieve fast and accurate tracking control of a quadrotor subjected to external disturbance. Inspiration is drawn from a fuzzy neural network that is constructed by fuzzy logic and zeroing neural network (ZNN). Distinct from most sliding mode control approaches, two nonsingular sliding mode surfaces are formulated by employing general ZNN approaches and differentiable predefined-time activation functions. Furthermore, for the compensation of external disturbance, a dynamic adaptive parameter and a fuzzy adaptive parameter are designed in the attitude control law. The fuzzy adaptive parameter, generated by the Takagi–Sugeno fuzzy logic system, is incorporated to enhance the robustness while reducing the chattering phenomena resulting from the discontinuous sign function. Theoretical proofs are provided to demonstrate the predefined-time convergence and robustness of the closed-loop system. Finally, two trajectory tracking examples are offered to validate the convergence, robustness, and low-chattering characteristics of the closed-loop system under the developed ARPTNSMC scheme.

With high wind speed, low turbulence, and high output, offshore wind power has gradually become a new area of wind power development. Nevertheless, the chaos phenomenon of offshore wind turbines manifests in some severe environments and the entire power generation system is affected. A variety of projective synchronization schemes are proposed to control this phenomenon, but the research on fixed-time projective synchronization of offshore wind turbine systems (OWTSs) is scant and has flaws in robustness. Motivated by zeroing neural dynamics (ZND) with fixed-time convergence, this article presents a fixed-time robust controller (FXTRC) based on ZND for the projective synchronization of OWTSs. In design process, a novel activation function is constructed to guarantee the projective synchronization speed and enhance the robustness. It is rigorously calculated that the upper bound of the projective synchronization time is only related to system parameters. Furthermore, the projective synchronization progress of OWTSs under the FXTRC displays better robustness compared with other controllers, which is proven by simulation results.

A dynamic gain fixed-time (FXT) robust zeroing neural network (DFTRZNN) model is proposed to effectively solve time-variant equality constrained quaternion least squares problem (TV-EQLS). The proposed approach surmounts the shortcomings of conventional numerical algorithms which fail to address time-variant problems. The DFTRZNN model is constructed with a novel dynamic gain parameter and a novel activation function (NAF), which differs from previous zeroing neural network (ZNN) models. Moreover, the comprehensive theoretical derivation of the FXT stability and robustness of the DFTRZNN model is presented in detail. Simulation results further confirm the availability and superiority of the DFTRZNN model for solving TV-EQLS. Finally, the consensus protocols of multiagent systems are presented by utilizing the design scheme of the DFTRZNN model, which further demonstrates its practical application value.

As an extension of the Lyapunov equation, the time-varying plural Lyapunov tensor equation (TV-PLTE) can carry multidimensional data, which can be solved by zeroing neural network (ZNN) models effectively. However, existing ZNN models only focus on time-varying equations in field of real number. Besides, the upper bound of the settling time depends on the value of ZNN model parameters, which is a conservative estimation for existing ZNN models. Therefore, this article proposes a novel design formula for converting the upper bound of the settling time into an independent and directly modifiable prior parameter. On this basis, we design two new ZNN models called strong predefined-time convergence ZNN (SPTC-ZNN) and fast predefined (FP)-time convergence ZNN (FPTC-ZNN) models. The SPTC-ZNN model has a nonconservative upper bound of the settling time, and the FPTC-ZNN model has excellent convergence performance. The upper bound of the settling time and robustness of the SPTC-ZNN and FPTC-ZNN models are verified by theoretical analyses. Then, the effect of noise on the upper bound of settling time is discussed. The simulation results show that the SPTC-ZNN and FPTC-ZNN models have better comprehensive performance than existing ZNN models.

Time-varying complex-valued tensor inverse (TVCTI) is a public problem worthy of being studied, while numerical solutions for the TVCTI are not effective enough. This work aims to find the accurate solution to the TVCTI using zeroing neural network (ZNN), which is an effective tool in terms of solving time-varying problems and is improved in this article to solve the TVCTI problem for the first time. Based on the design idea of ZNN, an error-adaptive dynamic parameter and a new enhanced segmented signum exponential activation function (ESS-EAF) are first designed and applied to the ZNN. Then a dynamic-varying parameter-enhanced ZNN (DVPEZNN) model is proposed to solve the TVCTI problem. The convergence and robustness of the DVPEZNN model are theoretically analyzed and discussed. In order to highlight better convergence and robustness of the DVPEZNN model, it is compared with four varying-parameter ZNN models in the illustrative example. The results show that the DVPEZNN model has better convergence and robustness than the other four ZNN models in different situations. In addition, the state solution sequence generated by the DVPEZNN model in the process of solving the TVCTI cooperates with the chaotic system and deoxyribonucleic acid (DNA) coding rules to obtain the chaotic-ZNN-DNA (CZD) image encryption algorithm, which can encrypt and decrypt images with good performance.

In this article, a low-order zeroing neural network (LZNN), a high-order ZNN (HZNN), and a variable-parameter ZNN (VZNN) are designed and applied to the time-changing Cholesky decomposition of any positive-definite matrix, where the LZNN and HZNN models are generated based on the traditional and high-order evolutionary formulas, respectively. In addition, a new activation function (N-Acf) is applied to the LZNN, HZNN, and VZNN models to improve the convergence and robustness. Importantly, the LZNN and HZNN models activated by the N-Acf have faster predefined-time convergence velocity when solving the time-changing Cholesky decomposition problem of any positive-definite matrix, which is demonstrated via theoretical analysis and numerical experiments. Finally, in light of empirical and theoretical evidence, it can be established that the solution model of the VZNN model is able to undergo convergence to the theoretical solution of Cholesky decomposition despite the presence of interposing noise.

Image fusion can obtain the superior information and reduce the noise in the source image by designing a specific scheme. However, the noise in image fusion has been a difficult issue and hard to handle. In this article, a variable-gain fixed-time convergent and robust zeroing neural network (VFCR-ZNN) model is proposed to figure out the image fusion problem and the corresponding quadratic programming (QP) problem. In contrast to the original zeroing neural network model, the VFCR-ZNN model adopts a novel fixed-time activation function and a useful variable-gain parameter, which allows the VFCR-ZNN model to converge faster in fixed-time and realize noise immunity under external disturbance. The detailed theory is provided to support this point. Different numerical QP comparative examples are carried out to effectively corroborate the rightness of the theoretical analyses and the excellence of the VFCR-ZNN model. Additionally, the quality of fused images acquired by the VFCR-ZNN model is higher compared to existing state-of-the-art models for image fusion. Furthermore, the VFCR-ZNN model is successfully utilized in the repetitive motion of six-link robot manipulator to demonstrate its significant practical implications.

Time-varying matrix inversion (TVMI) is a basic mathematical problem, which is widely involved in many scientific fields. In this article, an event-triggered control fuzzy adaptive zeroing neural network (ETC-FAZNN) model is proposed for solving the TVMI problem, where the fuzzy adaptive convergence parameter (FACP) is got by the redesigned fuzzy logic system, which makes the ETC-FAZNN model adaptive. Meanwhile, the event-triggered control is introduced to control the update of the FACP, which improves the calculation speed of the ETC-FAZNN model. Moreover, a novel activation function called segmented predefined-time activation function is put forward in this article to improve the convergence and robustness of the ETC-FAZNN model. Theoretical analysis and simulation experiments reveal that the ETC-FAZNN model can realize stability, predefined-time convergence, robustness, and adaptability performances in solving the TVMI problem.

Two nonlinear zeroing neural network (ZNN) models with prescribed-time convergence for time-dependent matrix LR and QR factorization are proposed in this article. To do so, two algorithms and two error functions are constructed to transform the time-dependent matrix LR and QR factorization problems into time-dependent linear equation systems, respectively. Simultaneously, a new activation function is introduced based on the initial ZNN models for the prescribed-time convergence of models. The excellent performance (robustness and convergence) of the two proposed ZNN models are analyzed theoretically. Furthermore, the prescribed-time convergence and antinoise abilities of the proposed ZNN models are well demonstrated in numerical experiments. Finally, the proposed ZNN model is applied to the moving target location problem, and the results show that the location error is at the millimeter level.

As a category of the recurrent neural network (RNN), zeroing neural network (ZNN) can effectively handle time-variant optimization issues. Compared with the fixed-parameter ZNN that needs to be adjusted frequently to achieve good performance, the conventional variable-parameter ZNN (VPZNN) does not require frequent adjustment, but its variable parameter will tend to infinity as time grows. Besides, the existing noise-tolerant ZNN model is not good enough to deal with time-varying noise. Therefore, a new-type segmented VPZNN (SVPZNN) for handling the dynamic quadratic minimization issue (DQMI) is presented in this work. Unlike the previous ZNNs, the SVPZNN includes an integral term and a nonlinear activation function, in addition to two specially constructed time-varying piecewise parameters. This structure keeps the time-varying parameters stable and makes the model have strong noise tolerance capability. Besides, theoretical analysis on SVPZNN is proposed to determine the upper bound of convergence time in the absence or presence of noise interference. Numerical simulations verify that SVPZNN has shorter convergence time and better robustness than existing ZNN models when handling DQMI.

Motivated from the convergence capability achieved by gradient neural network (GNN) and zeroing neural network (ZNN) for matrix inversion, in this article, a novel hybrid GNN-ZNN (H-GNN-ZNN) model is proposed by introducing a fuzzy adaptive control strategy to generate a fuzzy adaptive factor that can change its size adaptively according to the residual error. Due to its fuzzy adaptability, this novel model is called the fuzzy adaptive GNN-ZNN (FA-GNN-ZNN) model for presentation convenience. We prove that the FA-GNN-ZNN model has the better performance than the existing H-GNN-ZNN model under the same conditions. In addition, different activation functions are applied to the FA-GNN-ZNN model to improve its performance further, and the corresponding theoretical analysis is given. Finally, comparative simulation results demonstrate the validity and superiority of the FA-GNN-ZNN model for matrix inversion.

This article, for the first time, extends the zeroing neural network (ZNN) method to address the problem of dynamic quaternion-valued matrix inversion. Due to the noncommutative property of quaternion multiplication, the complex representation method is first adopted to transform quaternion-valued matrices into the corresponding complex-valued matrices. Then, based on two kinds of ways to deal with nonlinear activation functions in the complex-valued domain, this article proposes two quaternion-valued ZNN (QVZNN) models for dynamic quaternion-valued matrix inversion. In addition, a novel nonlinear activation function is given to accelerate the convergence rate of the models to reach the predefined-time convergence. The detailed theoretical analysis, together with four theorems, are given to show the excellent properties of the QVZNN models. Furthermore, the upper bound of the convergence time is derived analytically with the residual error being zero theoretically. Finally, two numerical examples are provided to verify the theoretical results and the effectiveness of the QVZNN models for the dynamic quaternion-valued matrix inversion, and an application to mobile manipulator control is provided to indicate the practical application value of the QVZNN models.

Matrix inversion frequently occurs in the fields of science, engineering, and related fields. Numerous matrix inversion schemes are often based on the premise that the solution procedure is ideal and noise-free. However, external interference is generally ubiquitous and unavoidable in practice. Therefore, an integrated-enhanced zeroing neural network (IEZNN) model has been proposed to handle the time-variant matrix inversion issue interfered with by noise. However, the IEZNN model can only deal with small time-variant noise interference. With slightly larger noise interference, the IEZNN model may not converge to the theoretical solution exactly. Therefore, a variable-parameter noise-tolerant zeroing neural network (VPNTZNN) model is proposed to overcome shortcomings and improve the inadequacy. Moreover, the excellent convergence and robustness of the VPNTZNN model are rigorously analyzed and proven. Finally, compared with the original zeroing neural network (OZNN) model and the IEZNN model for matrix inversion, numerical simulations and a practical application reveal that the proposed VPNTZNN model has the best robust property under the same external noise interference.

In this article, a noise-suppression variable-parameter zeroing neural network (NSVPZNN) is proposed to handle the dynamic Sylvester equation. Differing from the previous zeroing neural networks (ZNNs), a new nonlinear activation function and an especially constructed time-variant parameter are developed to construct the novel NSVPZNN model. Therefore, the NSVPZNN model can achieve faster predefined-time convergence without noise disturbance and have stronger robust performance under multiple noises. Furthermore, the convergence upper bound of the NSVPZNN model is theoretically calculated, and a detailed proof of guaranteeing noise-tolerance performance is given. Numerical simulations verify that the NSVPZNN has better performance than the ZNN, the finite-time convergence ZNN model, the predefined-time convergence ZNN model, and the other variable-parameter ZNN when handling the dynamic Sylvester equation. Finally, the design method of the NSVPZNN is applied to the wheeled manipulator for tracking the butterfly trajectory, which further illustrates the model's reliability.

For solving complex-valued linear matrix equations with time-varying coefficients (CV-LME-TVC) in the complex field, this article proposes a parameter-changing and complex-valued zeroing neural network (PC-CVZNN) model through integrating a new parameter-changing function. As compared to previous complex-valued zeroing neural networks (CVZNNs) with fixed parameters and existing parameter-changing functions, the PC-CVZNN model can achieve superior performance due to the accelerated role of the new parameter-changing function. In parts of theoretical analysis, we take advantage of Lyapunov methodology to prove that the proposed PC-CVZNN model can acquire the global and super-exponential convergence when the linear activation function is adopted, and even acquire super finite-time convergence when the new sign-bi-power activation function and its modified one are used. In parts of numerical comparison experiments, it is shown that the PC-CVZNN model possesses faster convergence rate than fixed-parameter CVZNN models and other analogy neural networks with parameter-changing function, when applied to finding the solution of CV-LME-TVC. Importantly, an application of the proposed method to the mobile manipulator control provides the potential practical value of the PC-CVZNN model in the industrial field.